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Beukers-like supercongruences for generalized Apéry numbers

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Abstract

For positive integers \(f_1,f_2,m,l\), the author and Chetry defined a generalization of Apéry numbers \(A(f_1,f_2,m,l,\lambda )\) given by

$$\begin{aligned} A(f_1,f_2,m,l,\lambda ):=\sum _{j=0}^{f_2}{f_1+j\atopwithdelims ()j}^m{f_2\atopwithdelims ()j}^l\lambda ^j. \end{aligned}$$

In this article, we prove certain Beukers-like supercongruences for these generalized Apéry numbers.

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References

  1. Ahlgren, S., Ono, K.: A Gaussian hypergeometric series evaluation and Apéry number supercongruences. J. Reine Angew. Math. 518, 187–212 (2000)

    MathSciNet  MATH  Google Scholar 

  2. Ahlgren, S.: Gaussian hypergeometric series and combinatorial congruences, symbolic computation, number theory, special functions, physics and combinatorics. Dev. Math. 4, 1–12 (2001)

    MATH  Google Scholar 

  3. Apéry, R.: Irrationalité de \(\zeta (2)\) and \(\zeta (3)\). Asterisque 61, 11–13 (1979)

    MATH  Google Scholar 

  4. Beukers, F.: Some congruences for Apéry numbers. J. Number Theory 21, 141–155 (1985)

    Article  MathSciNet  Google Scholar 

  5. Beukers, F.: Another congruence for the Apéry numbers. J. Number Theory 25, 201–210 (1987)

    Article  MathSciNet  Google Scholar 

  6. Chowla, S., Cowles, J., Cowles, M.: Congruence properties of Apéry numbers. J. Number Theory 12, 188–190 (1980)

    Article  MathSciNet  Google Scholar 

  7. Deines, A., Fuselier, J.G., Long, L., Swisher, H., Tu, F.: Hypergeometric series, truncated hypergeometric series, and Gaussian hypergeometric functions. In: Directions in number theory, vol. 3, Assoc. Women Math. Ser., pp. 125–159. Springer, Cham (2016)

    MATH  Google Scholar 

  8. Gessel, I.: Some congruences for Apéry numbers. J. Number Theory 14, 362–368 (1982)

    Article  MathSciNet  Google Scholar 

  9. Greene, J.: Hypergeometric functions over finite fields. Trans. Am. Math. Soc. 301(1), 77–101 (1987)

    Article  MathSciNet  Google Scholar 

  10. Ishikawa, T.: Supercongruence for the Apéry numbers. Nagoya Math. J. 118, 195–202 (1990)

    Article  MathSciNet  Google Scholar 

  11. Kalita, G., Chetry, A.S.: Congruences for generalized Apéry numbers and Gaussian hypergeometric series. Res. Number Theory 3, 15 (2017)

    Article  Google Scholar 

  12. Kilbourn, T.: An extension of the Apéry number supercongruence. Acta Arith. 123(4), 335–348 (2006)

    Article  MathSciNet  Google Scholar 

  13. Koblitz, N.: p-Adic analysis: a short course on recent work. Cambridge University Press, Cambridge (1980)

    Book  Google Scholar 

  14. Koike, M.: Hypergeometric series over finite fields and Apéry numbers. Hiroshima Math. J. 22, 461–467 (1992)

    MathSciNet  MATH  Google Scholar 

  15. McCarthy, D., Osburn, R.: A p-adic analogue of a formula of Ramanujan. Arch. Math. 91(6), 492–504 (2008)

    Article  MathSciNet  Google Scholar 

  16. Mimura, Y.: Congruence properties of Apéry numbers. J. Number Theory 16, 138–146 (1983)

    Article  MathSciNet  Google Scholar 

  17. Mortenson, E.: A supercongruence conjecture of Rodriguez-Villegas for a certain truncated hypergeometric function. J. Number Theory 99(1), 139–147 (2003)

    Article  MathSciNet  Google Scholar 

  18. Ono, K.: Values of Gaussian hypergeometric series. Trans. Am. Math. Soc. 350(3), 1205–1223 (1998)

    Article  MathSciNet  Google Scholar 

  19. Radoux, C.: Quelques propriétés arithmétiques des nombres d’Apéry. C. R. Acad. Sci. Paris A291, 567–569 (1980)

    MATH  Google Scholar 

  20. Rodriguez-Villegas, F.: Hypergeometric families of Calabi–Yau manifolds. Am. Math. Soc. 38, 223–243 (2003)

    MathSciNet  MATH  Google Scholar 

  21. Stienstra, J., Beukers, F.: On the Picard–Fuchs equation and the formal Brauer group of certain elliptic K3-surfaces. Math. Ann. 271(2), 269–304 (1985)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

We thank Ken Ono for going through the initial draft of the paper and many helpful suggestions. We are grateful to the referee for his/her helpful comments.

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Authors and Affiliations

  1. Department of Sciences and Mathematics, Indian Institute of Information Technology Guwahati, Ambari, GNB Road, Assam, 781001, India

    Gautam Kalita

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  1. Gautam Kalita
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Kalita, G. Beukers-like supercongruences for generalized Apéry numbers. Ramanujan J 47, 501–508 (2018). https://doi.org/10.1007/s11139-017-9932-3

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  • DOI: https://doi.org/10.1007/s11139-017-9932-3

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